Attribute significance is very important in multiple-attribute decision-making (MADM) problems. In a MADM problem, the significance of attributes is often different. In order to overcome the shortcoming that attribute significance is usually given artificially. The purpose of this paper is to give attribute significance computation formulas based on inclusion degree. We note that in the real-world application, there is a lot of incomplete information due to the error of data measurement, the limitation of data understanding and data acquisition, etc. Firstly, we give a general description and the definition of incomplete information systems. We then establish the tolerance relation for incomplete linguistic information system, with the tolerance classes and inclusion degree, significance of attribute is proposed and the corresponding computation formula is obtained. Subsequently, for incomplete fuzzy information system and incomplete interval-valued fuzzy information system, the dominance relation and interval dominance relation is established, respectively. And the dominance class and interval dominance class of an element are got as well. With the help of inclusion degree, the computation formulas of attribute significance for incomplete fuzzy information system and incomplete interval-valued fuzzy information system are also obtained. At the same time, results show that the reduction of attribute set can be obtained by computing the significance of attributes in these incomplete information systems. Finally, as the applications of attribute significance, the attribute significance is viewed as attribute weights to solve MADM problems and the corresponding TOPSIS methods for three incomplete information systems are proposed. The numerical examples are also employed to illustrate the feasibility and effectiveness of the proposed approaches.
Abo-SinnaM.A. and AmerA.H., Extensions of TOPSIS formulti-objective large-scale nonlinear programming problems, Applied Mathematics and Computation162 (2005), 243–256.
2.
AhnaB.S., ParkbK.S., HanbC.H. and KimJ.K., Multiattribute decision aid under incomplete information and hierarchical structure, European Journal of Operational Research125(2000), 431–439.
3.
AydinaS., KahramanbC. and KayaI., A new fuzzy multicriteria decision making approach: An application for European Quality Award assessment, Knowledge-Based Systems32 (2012), 37–46.
4.
BaşhanV., DemirelH. and GulM., An FMEA-based TOPSIS approach under single valued neutrosophic sets for maritime risk evaluation: the case of ship navigation safety, Soft Computing24(2020), 18749–18764.
5.
BaşhanV., DemirelH. and GulM., A novel risk evaluation approach for frequently encountered risks in ship engine rooms, Brodogradnja71 (2020), 31–54.
6.
ChinK.S., FuC. and WangY.M., A method of determining attribute weights in evidential reasoning approach based on in compatibility among attributes, Computers and Industrial Engineering87 (2015), 150–162.
7.
ChuT.C. and LinY.C., An interval arithmetic based fuzzy TOPSIS model, Expert Systems with Applications36 (2009), 10870–10876.
8.
DaiJ.H., WeiB.J., ZhangX.H. and ZhangQ.L., Uncertainty measurement for incomplete interval-valued information systems based on α-weak similarity, Knowledge-Based Systems136 (2017), 159–171.
9.
DembczyńskiK., GrecoS. and SlowińskiR., Rough set approach to multiple criteria classification with imprecise evaluations and assignments, European Journal of OperationalResearch198 (2009), 626–636.
10.
GargH. and AroraR., TOPSIS method based on correlation coefficient for solving decision-making problems with intuitionistic fuzzy soft set information, AIMS Mathematics5 (2020), 2944–2966.
11.
GargH. and KaurG., TOPSIS based on nonlinear programming methodology for solving decision-making problems under cubic intuitionistic fuzzy set environment, Computational and Applied Mathematics38 (2019), 114–133.
12.
GargH. and KumarK., A novel exponential distance and its based TOPSIS method for interval-valued intuitionistic fuzzy sets using connection number of SPA theory, Artificial Intelligence Review53 (2020), 595–624.
13.
GrecoS., MatarazzoB. and SlowinskiR., Rough sets theory for multicriteria decision analysis, European Journal of Operational Research129 (2001), 1–47.
14.
Grzymala-BusseJ.W., Characteristic relations for incomplete data: a generalization of the indiscernibility relation, in: S. Tsumoto et al. (Eds.), Proceeding of the Third International Conference on Rough Sets and Current Trends in Computing, Lecture Notes in Artificial Intelligence, vol. 3066, Springer-Verlag, Berlin, Heidelberg, New York, (2004), 244–253.
15.
GulM. and AkM.F., A modified failure modes and effects analysisusing interval-valued spherical fuzzy extension of TOPSIS method: case study in a marble manufacturing facility, Soft Computing25 (2021), 6157–6178.
16.
GuoK.H. and LiW.L., An attitudinal-based method for constructing intuitionistic fuzzy information in hybrid MADM under uncertainty, Information Sciences208 (2012), 28–38.
17.
HuaZ.S., GongB.G. and XuX.Y., A DS-AHP approach formulti-attribute decision making problem with incomplete information, Expert Systems with Applications34 (2008), 2221–2227.
18.
HwangC.L. and YoonK., Multiple attribute decision making: methods and applications, Springer, Berlin, Heidelberg, New York, (1981), 78–98.
19.
JahanshahlooG.R., Hosseinzadeh LotfiF. and IzadikhahM., An algorithmic method to extend TOPSIS for decision-making problems with interval data, Applied Mathematics and Computation175 (2006), 1375–1384.
20.
JahanshahlooG.R., Hosseinzadeh LotfiF. and DavoodiA.R., Extension of TOPSIS for decision-making problems with interval data: interval efficiency, Mathematical and Computer Modelling49(2009), 1137–1142.
21.
JenaR. and PradhanB., Integrated ANN-cross-validation and AHP-TOPSIS model to improve earthquake risk assessment, International Journal of Disaster Risk Reduction50 (2020), 101723.1–101723.16.
22.
KacprzakD., A doubly extended TOPSIS method for group decision making based on ordered fuzzy numbers, Expert Systems with Applications116 (2019), 243–254.
23.
KeikhaA., GargH. and NehiH.M., An approach based on combining Choquet integral and TOPSIS methods to uncertain MAGDM problems, Soft Computing25 (2021), 7181–7195.
24.
KryszkiewiczM., Rough set approach to incomplete information systems, Information Sciences112 (1998), 39–49.
25.
KryszkiewiczM., Rules in incomplete information systems, Information Sciences113 (1999), 271–292.
26.
KumarA. and AnbanandamR., Location selection of multimodal freight terminal under STEEP sustainability, Research in Transportation Business and Management33 (2019), 100434.1–100434.16.
27.
KuoR.J., WuY.H. and HsuT.S., Integration of fuzzy set theory and TOPSIS into HFMEA to improve outpatient service for elderly patientsin Taiwan, Journal of the Chinese Medical Association75(2012), 341–348.
28.
LatkowskiR., Flexible in discernibility relations for missing attribute values, Fundamenta Informaticae67 (2005), 131–147.
29.
LeeJ.W., ImH.S. and YouG.W., Optimizing skyline queries over incomplete data, Information Sciences361 (2016), 14–28.
30.
LeungY., WuW.Z. and ZhangW.X., Knowledge acquisition in incomplete information systems: a rough set approach, European Journal of Operational Research168 (2006), 164–180.
31.
LiangJ.Y., ShiZ.Z. and LiD.Y., Information entropy, rough entropyand knowledge granulation in incomplete information systems, International Journal of General Systems35 (2006), 641–654.
32.
LiangD.C., HeX. and XuZ.S., Multi-attribute dynamic two sided matching method of talent sharing market in incomplete preference ordinal environment, Applied Soft Computing93 (2020), 106427.1–106427.17.
33.
LipskiW.J., On semantic issues connected with incomplete information databases, ACM Transactions on Databases Systems4 (1979), 262–296.
34.
LiuD., LiangD.C. and WangC.C., A novel three-way decision model based on incomplete information system, Knowledge-BasedSystems91 (2016), 32–45.
35.
LuM.T., TsaiJ.F., ShenS.P., LinM.H. and HuY.C., Estimating sustainable development performance in the electrical wire and cableindustry: Applying the integrated fuzzy MADM approach, Journalof Cleaner Production277 (2020), 122440.
36.
PalczewskiK. and SalabunW., The fuzzy TOPSIS applications in the last decade, Procedia Computer Science159 (2019), 2294–2303.
37.
PawlakZ., Rough sets, International Journal of Computer and Information Sciences11 (1982), 314–356.
38.
QianW.B. and ShuW.H., Attribute reduction in incomplete ordered information systems with fuzzy decision, Applied Soft Computing73 (2018), 242–253.
SunB.Z., QiC., MaW.M., WangT., ZhangL.Y. and JiangC., Variable precision diversified attribute multigranulation fuzzy roughset-based multi-attribute group decision making problems, Computers and Industrial Engineering142 (2020), 106331.1–106331.15.
41.
WangL., GargH. and LiN., Pythagorean fuzzy interactive Hamacher power aggregation operators for assessment of express servicequality with entropy weight, Soft Computing25 (2021), 973–993.
42.
WuX.C., Existence of value for differential games with incomplete information and signals on initial states and payoffs, Journal of Mathematical Analysis and Applications446 (2017), 1196–1218.
43.
XuZ.S. and ZhangX.L., Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information information, Knowledge-Based Systems52 (2013), 53–64.
44.
YangX.B., YuD.J., YangJ.Y. and WeiL.H., Dominance-based roughset approach to incomplete interval-valued information system, Data and Knowledge Engineering68 (2009), 1331–1347.
45.
YangX.B., ZhangM., DouH.L. and YangJ.Y., Neighborhood systems-based rough sets in incomplete information system, Knowledge-Based Systems24 (2011), 858–867.
46.
YeF., An extended TOPSIS method with interval-valued intuitionistic fuzzy numbers for virtual enterprise partner selection, Expert Systems with Applications37 (2010), 7050–7055.
47.
YeF. and LiY.N., Group multi-attribute decision model to partner selection in the formation of virtual enterprise under incomplete information, Expert Systems with Applications36 (2009), 9350–9357.
48.
YucesanM. and GulM., Hospital service quality evaluation: anintegrated model based on Pythagorean fuzzy AHP and fuzzy TOPSIS, Soft Computing24 (2020), 3237–3255.
49.
ZhangK., ZhanJ.M. and WangX.Z., TOPSIS-WAA method based on a covering-based fuzzy rough set: An application to rating problem, Information Sciences539 (2020), 397–421.
50.
ZhangL., ZhanJ.M. and YaoY.Y., Intuitionistic fuzzy TOPSIS method based on CVPIFRS models: An application to biomedical problems, Information Sciences517 (2020), 315–339.
51.
ZhangZ.X., LiangH.J., MaH. and PanY.N., Reliable fuzzy control for uncertain vehicle suspension systems with random incomplete transmission signals and sensor failure, Mechanical Systems and Signal Processing130 (2019), 776–789.
